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Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

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  • Florios, Kostas
  • Mavrotas, George
  • Diakoulaki, Danae

Abstract

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive [epsilon]-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932-942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive [epsilon] -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).

Suggested Citation

  • Florios, Kostas & Mavrotas, George & Diakoulaki, Danae, 2010. "Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms," European Journal of Operational Research, Elsevier, vol. 203(1), pages 14-21, May.
    • Handle: RePEc:eee:ejores:v:203:y:2010:i:1:p:14-21
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    1. Jaszkiewicz, Andrzej, 2004. "On the computational efficiency of multiple objective metaheuristics. The knapsack problem case study," European Journal of Operational Research, Elsevier, vol. 158(2), pages 418-433, October.
    2. Thomas Erlebach & Hans Kellerer & Ulrich Pferschy, 2002. "Approximating Multiobjective Knapsack Problems," Management Science, INFORMS, vol. 48(12), pages 1603-1612, December.
    3. Laumanns, Marco & Thiele, Lothar & Zitzler, Eckart, 2006. "An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method," European Journal of Operational Research, Elsevier, vol. 169(3), pages 932-942, March.
    4. Matthias Ehrgott, 2006. "A discussion of scalarization techniques for multiple objective integer programming," Annals of Operations Research, Springer, vol. 147(1), pages 343-360, October.
    5. Gomes da Silva, Carlos & Climaco, Joao & Figueira, Jose, 2006. "A scatter search method for bi-criteria {0, 1}-knapsack problems," European Journal of Operational Research, Elsevier, vol. 169(2), pages 373-391, March.
    6. Shukla, Pradyumn Kumar & Deb, Kalyanmoy, 2007. "On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1630-1652, September.
    7. Bazgan, Cristina & Hugot, Hadrien & Vanderpooten, Daniel, 2009. "Implementing an efficient fptas for the 0-1 multi-objective knapsack problem," European Journal of Operational Research, Elsevier, vol. 198(1), pages 47-56, October.
    8. Alves, Maria Joao & Climaco, Joao, 2007. "A review of interactive methods for multiobjective integer and mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 180(1), pages 99-115, July.
    9. Mavrotas, G. & Diakoulaki, D., 1998. "A branch and bound algorithm for mixed zero-one multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 107(3), pages 530-541, June.
    10. Zhang, Cai Wen & Ong, Hoon Liong, 2004. "Solving the biobjective zero-one knapsack problem by an efficient LP-based heuristic," European Journal of Operational Research, Elsevier, vol. 159(3), pages 545-557, December.
    11. Gomes da Silva, Carlos & Figueira, Jose & Climaco, Joao, 2007. "Integrating partial optimization with scatter search for solving bi-criteria {0, 1}-knapsack problems," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1656-1677, March.
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